Let $z_1,\dots ,z_n\in \mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of$$s_k=\sum _{1\le i\le n}{z}_i^k$$contains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ which are all $0$. Then (essentially)$$z_j=e(j/n),$$where $e(x)=e^{2\pi ix}$.